Dependence or Independence of the Sample Mean and Variance In Non-IID or Non-Normal Cases: How Helpful Are Some of the Customary Tests of Independence?

• Activity Number: 149
• Type: Contributed
• Date/Time: Monday, August 12, 2002 : 2:00 PM to 3:50 PM
• Sponsor: IMS
• Author(s): Nitis Mukhopadhyay*+
• Affiliation(s): University of Connecticut
• Keywords: tests for independence ; correlation test ; P-value

Abstract

It is well-known (Cramer,1946) that if the underlying common probability model for each observation in a random sample of size n is normal, then the sample mean and sample variance are independently distributed. On the other hand, it is also known that if the sample mean and variance are independently distributed, then the underlying common probability model for each observation must be normal (Zinger, 1958) when the observations are independent and identically distributed (iid). But, what can one expect regarding the status of independence or dependence between the sample mean and variance when the observations are allowed to be non-iid or non-normal? With the help of examples, we highlight a number of interesting possibilities. Illustrations through simulated data are provided where we have applied the customary t-test based on Pearson-sample correlation coefficient, a traditional non-parametric test based on Spearman-rank correlation coefficient, and the Chi-square test to "validate" independence or dependence between the appropriate variables (the sample mean and variance) under consideration. We find that among three contenders, the Chi-square test is most consistent.